Rad HAZU, Matematičke znanosti, Vol. 18 (2014), 73-89.
GENERALIZATIONS AND IMPROVEMENTS OF AN INEQUALITY OF HARDY-LITTLEWOOD-PÓLYA
Sadia Khalid and Josip Pečarić
Abdus Salam School of Mathematical Sciences, GC University, 68-B, New Muslim Town,
Lahore 54600, Pakistan
Faculty of Textile Technology, University of Zagreb, Prilaz baruna Filipovića 28a
10000 Zagreb, Croatia
Abstract. Some generalizations of an inequality of
Hardy-Littlewood-Pólya are presented. We discuss the n-exponential convexity
and log-convexity of the functions associated with the linear functional
defined by the generalized inequality and also prove the monotonicity
property of the generalized Cauchy means obtained via this functional. Finally,
we give several examples of the families of functions for which the results
can be applied.
2010 Mathematics Subject Classification.
26A48, 26A51, 26D15.
Key words and phrases. Non-increasing sequence in mean, Wright-convex function,
n-exponential and logarithmic convexity, divided difference.
Full text (PDF) (free access)
- H. D. Brunk,
Integral inequalities for functions with non-decreasing increments,
Pacific J. Math 14 (1964), 783-793.
- G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, 2nd ed.,
Cambridge Univ. Press, Cambridge, 1952.
- J. Jakšetić and J. Pečarić,
Exponential Convexity Method,
J. Convex Anal. 20 (2013), 181-197.
- C. Jardas, J. Pečarić, R. Roki and N. Sarapa,
On some inequalities for entropies of discrete probability distributions,
J. Austral. Math. Soc. Ser. B 40 (1999), 535-541.
- J. Pečarić and L. E. Persson,
On an inequality of Hardy-Littlewood-Pólya, Math. Gazette 79 (1995), 383-385.
- J. Pečarić and J. Perić,
Improvements of the Giaccardi and the Petrović inequality and related Stolarsky type means,
An. Univ. Craiova Ser. Mat. Inform. 39 (2012), 65-75.
- J. Pečarić, F. Proschan and Y. L. Tong,
Convex Functions, Partial Orderings, and Statistical Applications,
Academic Press Inc., 1992.
- D. V. Widder,
The Laplace Transform.
Princeton Univ. Press, New Jersey, 1941.
Rad HAZU Home Page